Writing Linear Functions

Transcription

1 2- Writing Linear Functions Objectives Use slope-intercept form and point-slope form to write linear functions. Write linear functions to solve problems. Vocabulary Point-slope form Why learn this? When you play Monopoly, it s easy to calculate the rent of most properties by looking at the selling price. (See Example.) Recall from Lesson 2- that the slopeintercept form of a linear equation is, where m is the slope of the line and b is its y-intercept. In Lesson 2-, you graphed lines when you were given the slope and y-intercept. In this lesson you will write linear functions when you are given graphs of lines or problems that can be modeled with a linear function. EXAMPLE 1 Writing the Slope-Intercept Form of the Equation of a Line To express a line as a linear function, replace y with f (x). y = - 2 _ 5 x + 2 f (x) = - 2 _ 5 x + 2 Write the equation of the graphed line in slope-intercept form. Step 1 Step 2 Identify the y-intercept. The y-intercept b is 2. Find the slope. Choose any two convenient points on the line, such as (0, 2) and (5, 0). Count from (0, 2) to (5, 0) to find the rise and the run. The rise is 2 units and the run is 5 units. Slope is _ rise run = _ -2 =-_ Step Write the equation in slope-intercept form. 2_ y = - 5 x + 2 m = - _ 2 5 and b = 2 The equation of the line is y =-_ 2 x Write the equation of the graphed line in slopeintercept form. Notice that for two points on a line, the rise is the difference in the y-coordinates, and the run is the difference in the x-coordinates. Using this information, we can define the slope of a line by using a formula. 2- Writing Linear Functions 115

2 Slope Formula WORDS ALGEBRA GRAPH Given two points on a line, the slope is the ratio of the difference in the y-values to the difference in the corresponding x-values, or rise over run. The slope of the line containing ( x 1, y 1 ) and ( x 2, y 2 ) is x 2 - x. 1 EXAMPLE 2 Finding the Slope of a Line Given Two or More Points Find the slope of each line. A the line through (, 2) and ( 1, 2) Let be ( x 1, y 1 ) be (, 2) and ( x 2, y 2 ) be ( 1, 2). x 2 - x = _ 2 - (-2) = _ - The slope of the line is 1. = -1 Use the slope formula. If you reverse the order of the points in Example 2B, the slope is still the same. m = _ = _ - -6 = _ 5 B x y Let ( x 1, y 1 ) be ( 5, 6), and ( x 2, y 2 ) be (11, 16). x 2 - x = _ = _ 6 = _ 5 The slope of the line is 5 _. Choose any two points. Use the slope formula. C The line shown. Either point may be chosen as ( x 1, y 1 ). Let ( x 1, y 1 ) be (2, 1) and ( x 2, y 2 ) be (2, ). x 2 - x = _ - (-1) = _ 0 Because division by zero is undefined, the slope of the line is undefined. Find the slope of each line. 2a. x y b. the line through (2, -5) and (-, -5) Because the slope of a line is constant, it is possible to use any point on a line and the slope of the line to write an equation of the line in point-slope form. 116 Chapter 2 Linear Functions

3 Point-Slope Form The equation of a line with a slope of m and the point ( x 1, y 1 ) is y - y 1 = m (x - x 1 ). EXAMPLE Writing Equations of Lines In slope-intercept form, write the equation of x the line that contains the points in the table. y First, find the slope. Let ( x 1, y 1 ) be (-1, 1) and ( x 2, y 2 ) be (, 0). x 2 - x = _ (-1) = _ = - Next, choose a point and use either form of the equation of a line. Method A Point-Slope Form Method B Slope-Intercept Form Using (, 0) : Using (, 0), solve for b. y - y 1 = m (x - x 1 ) y - (0) = - (x - ) Substitute. 0 = ( - ) + b Substitute. y = - (x - ) Simplify. 0 = - _ + b Simplify. Rewrite in slope-intercept form. b = _ Solve for b. y = - 1 _ (x - ) Rewrite the equation using m and b. y = - x + Distribute. y = - x + The equation of the line is y = - x + _. a. with slope 5 through (1, ) b. through (-2, -) and (2, 5) Point-Slope and Point-Slope Form Form I learned the FPO point-slope form by relating it to the formula for slope. The formula A207SE-C02L0-001P for slope and point-slope form are basically the same equation in different forms. Begin with the slope formula: m = y _ 2 - y 1 x 2 - x 1 Substitute (x, y) for ( x 2, y 2 ): m = y _ - y 1 x - x 1 Jennifer Chang Jefferson High School Multiply both sides by (x - x 1 ): m (x - x 1) = y - y 1 Reverse the equation: y - y 1 = m (x - x 1) 2- Writing Linear Functions 117

4 EXAMPLE Entertainment Application In the game of Monopoly, a player who lands on a property that is owned by another player must pay rent to the owner of the property. For most color properties, the rent can be modeled by a linear function of the selling price. A Express the rent as a function of the selling price. Let x = selling price and y = rent. Find the slope by choosing two points. Let (x 1, y 1) be (60, 2) and (x 2, y 2) be (0, 6). x 2 - x = _ = _ 0 = _ 1 To find the equation for the rent function, use point-slope form. y - y 1 = m (x - x 1 ) y - 2 = (x - 60 ) Use the data for Mediterranean Ave. _ y = 1 x - Simplify. B Graph the relationship between the selling price and the rent. How much is the rent for Illinois Ave., which has a selling price of $20? Graph the function using a scale that fits the data. To find the rent for Illinois Avenue, use the graph or substitute its selling price of $20 into the function. y = _ 1 (20) - Substitute. y = 2 - y = 20 The rent for Illinois Avenue is $20. a. Express the cost as a linear function of the number of items. b. Graph the relationship between the number of items and the cost. Find the cost of 18 items. Items Cost ($) Chapter 2 Linear Functions

5 By comparing slopes, you can determine if lines are parallel or perpendicular. You can also write equations of lines that meet certain criteria. Parallel and Perpendicular Lines WORDS GRAPH ALGEBRA Parallel Lines If both slopes are defined, the slopes of parallel lines are equal. The slopes of parallel vertical lines are undefined. y 1 = 2x + 1, so m 1 = 2 y 2 = 2x - so m 2 = 2 m 1 = m 2 2 = 2 A vertical line has an undefined slope. Perpendicular Lines If both slopes are defined, the slopes of perpendicular lines are opposite reciprocals. Their product is -1. A vertical line and a horizontal line are perpendicular. y 1 = - _ x +, so 2 m 1 = - _ 2 y 2 = 2_ 2_ x -, so m 2 = ( m 1 )( m 2 ) = -1 ( - _ ) = -1 2) ( 2 _ EXAMPLE 5 Writing Equations of Parallel and Perpendicular Lines A parallel to y = 1.5x + 6 and through (, 5) m = 1.5 Parallel lines have equal slopes. y - 5 = 1.5 (x - ) Use y - y 1 = m (x - x 1 ) with ( x 1, y 1 ) = (, 5). y - 5 = 1.5x - 6 y = 1.5x - 1 Distributive property. Simplify. B perpendicular to y = -_ x + 2 and through (6, -) The slope of the given line is -, so the slope of the perpendicular line is the opposite reciprocal,. y + = _ x - 6 Use y - y 1 = m (x - x 1 ). y + is equivalent to y -(-). y + = _ x - 8 Distributive property. y = _ x - 12 Simplify. 5a. parallel to y = 5x - and through (1, ) 5b. perpendicular to y = _ 5 x - 7 and through (0, -2) 6 2- Writing Linear Functions 119